Cailín An Tí Mhóir

Title Error

I’m no Gaelic scholar, but if the translation is to be understood as "the girl of the big house", the Gaeilge bit should read: "An Cailín den Teach Mhór". The only time the definite article ‘an’ (the) is used to imply ‘of the’ is when its plural alternate ‘na’ precedes the plural form of a noun: An Cailin na Tithe Mhor, the girls of the big houses.

I can think of plenty of examples in Irish of "an" being used to imply genitive "of the" in the singular. Some are "Fear a(n) Ti" = the man OF the house, also the name of a nice reel. Other examples and tune names are "barr an chnoic" = the top of the hill, "bean an tinceara" (The Tinker’s Wife), "bairead an mhairnealaigh" (The Sailor’s Bonnet). The song "bean an fhir rua" (the Red Haired Man’s Wife). Sorry can’t get the fadas. You may be getting confused with the use of preposition "de" with partitive genitive.